PlanetMath: direct sum of Hermitian and skew-Hermitian matrices

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direct sum of Hermitian and skew-Hermitian matrices

(Example)

In this example, we show that any square matrix with complex entries can uniquely be decomposed into the sum of one Hermitian matrix and one skew-Hermitian matrix. A fancy way to say this is that complex square matrices is the direct sum of Hermitian and skew-Hermitian matrices.

Let us denote the vector space (over $\mathbb{C}$ ) of complex square $n\times n$ matrices by . Further, we denote by respectively the vector subspaces of Hermitian and skew-Hermitian matrices. We claim that

$\displaystyle M$

$\displaystyle =$

$\displaystyle M_+\oplus M_-.$

(1)

Since

and

are vector subspaces of

, it is clear that

is a vector subspace of

. Conversely, suppose $A\in M$ . We can then define

$\displaystyle A_+$	$\displaystyle =$	$\displaystyle \frac{1}{2}\big( A + A^\ast \big),$
$\displaystyle A_-$	$\displaystyle =$	$\displaystyle \frac{1}{2}\big( A - A^\ast \big).$

Here $A^\ast = \overline{A}\hspace{0.04cm} ^{\mbox{\scriptsize {T}}} \hspace{0.02cm}$ , and $\overline{A}$ is the complex conjugate of

, and $A\hspace{0.04cm} ^{\mbox{\scriptsize {T}}} \hspace{0.02cm}$ is the transpose of

. It follows that

is Hermitian and

is anti-Hermitian. Since

, any element in

can be written as the sum of one element in

and one element in

. Let us check that this decomposition is unique. If $A\in M_+\cap M_-$ , then $A=A^\ast=-A$ , so

. We have established equation 1.

Special cases

In the special case of $1\times 1$ matrices, we obtain the decomposition of a complex number into its real and imaginary components.
In the special case of real matrices, we obtain the decomposition of a $n\times n$ matrix into a symmetric matrix and anti-symmetric matrix.

"direct sum of Hermitian and skew-Hermitian matrices" is owned by mathcam. [ full author list (2) | owner history (1) ]

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See Also: direct sum of even/odd functions (example)

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Cross-references: anti-symmetric, symmetric matrix, components, imaginary, real, equation, decomposition, transpose, complex conjugate, clear, vector subspaces, matrices, square, vector space, skew-Hermitian matrix, Hermitian matrix, sum, complex, square matrix
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This is version 2 of direct sum of Hermitian and skew-Hermitian matrices, born on 2003-05-03, modified 2006-06-20.
Object id is 4238, canonical name is DirectSumOfHermitianAndSkewHermitianMatrices.
Accessed 4523 times total.

Classification:

AMS MSC:	15A57 (Linear and multilinear algebra; matrix theory :: Other types of matrices )
	15A03 (Linear and multilinear algebra; matrix theory :: Vector spaces, linear dependence, rank)

Pending Errata and Addenda

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